Evaluate lim x→∞ (sqrt(x^2 + x) − x).

Evaluate lim x→∞ (sqrt(x^2 + x) − x).

• A. 1/2
• B. 0
• C. -1/2
• D. 1

Answer: (A)

When a limit involves a difference with a square root like sqrt(x^2 + x) − x, the best move is to rationalize by multiplying by the conjugate sqrt(x^2 + x) + x. This gives sqrt(x^2 + x) − x = [ (x^2 + x) − x^2 ] / [ sqrt(x^2 + x) + x ] = x / [ sqrt(x^2 + x) + x ]. Factor out x in the denominator by dividing numerator and denominator by x: = 1 / [ sqrt(1 + 1/x) + 1 ]. As x → ∞, sqrt(1 + 1/x) → 1, so the limit becomes 1 / (1 + 1) = 1/2. So the limit equals 1/2. The expression is positive for large x and approaches a finite positive value, not 0 or negative, and the denominator tends to 2 rather than 1, which is why the result is 1/2 rather than 0, −1/2, or 1.

When a limit involves a difference with a square root like sqrt(x^2 + x) − x, the best move is to rationalize by multiplying by the conjugate sqrt(x^2 + x) + x. This gives

sqrt(x^2 + x) − x = [ (x^2 + x) − x^2 ] / [ sqrt(x^2 + x) + x ] = x / [ sqrt(x^2 + x) + x ].

Factor out x in the denominator by dividing numerator and denominator by x:

= 1 / [ sqrt(1 + 1/x) + 1 ].

As x → ∞, sqrt(1 + 1/x) → 1, so the limit becomes 1 / (1 + 1) = 1/2.

So the limit equals 1/2. The expression is positive for large x and approaches a finite positive value, not 0 or negative, and the denominator tends to 2 rather than 1, which is why the result is 1/2 rather than 0, −1/2, or 1.